(************** Content-type: application/mathematica **************
                     CreatedBy='Mathematica 5.2'

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(*NotebookOptionsPosition[     88170,       2385]*)
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Notebook[{
Cell[BoxData[
    \(<< ACPackages`\)], "Input"],

Cell[BoxData[
    \(v[y_] := y \((H - y)\) 4/H\^2; H = 2;\)], "Input"],

Cell[BoxData[
    \(\(eqOZ[\[Omega]_, k_, 
          R_] := \((v[z] - \[Omega]\/k)\) 
            f[z] - \(\[ImaginaryI]\/\(k\ R\)\) \((\(f''\)[z] - \(k\^2\) 
                  f[z])\);\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \( (*\:041a\:0420\:0421\ \:0441\ \:0446\:0435\:043b\:043e\:0447\:0438\
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\:0439\ \:0438\ \:043e\:0434\:0438\:043d\:0430\:043a\:043e\:0432\:044b\:043c\
\:0438\ \:0440\:0430\:0437\:043d\:043e\:0441\:0442\:044f\:043c\:0438*) \)], \
"Input"],

Cell[BoxData[
    \(\(FiniteMatrix4Eigen[n_, sample_:  5, subst___] := 
        Block[{h = 1\/n, krseq, krs, unk, unkmain, unkfict, i, sm, s, order, 
            lhs, rhs, \[Omega]}, \[IndentingNewLine]sm = 
            Max[sample, 5]; \[IndentingNewLine]unk = 
            Table[f\_i, {i, 0 - \(sm - 1\)\/2, 
                n + \(sm - 1\)\/2}]; \[IndentingNewLine]unkmain = 
            Table[f\_i, {i, 1, n - 1}]; \[IndentingNewLine]unkfict = 
            Complement[unk, unkmain]; 
          dersam[order_, s_] := 
            Table[f\_j, {j, i - \(s - 1\)\/2, i + \(s - 1\)\/2}] . 
                NDCoefficientList[order, s]/
              h\^order; \[IndentingNewLine]rhs = \(\(\[Omega]\/k\) \
\((\(\(f'\)'\)[z] - \(k\^2\) f[z])\) /. {z \[Rule] i\ h, 
                  f[z] \[Rule] f\_i, \(f''\)[z] \[Rule] 
                    dersam[2, sm], \(f''''\)[z] \[Rule] 
                    dersam[4, 
                      sm]}\) /. {subst}; \[IndentingNewLine]lhs = \(Simplify[\
\((eqOZ[\[Omega], k, R] + \(\[Omega]\/k\) \((\(\(f'\)'\)[z] - \(k\^2\) 
                              f[z])\))\)] /. {z \[Rule] i\ h, 
                  f[z] \[Rule] f\_i, \(f''\)[z] \[Rule] 
                    dersam[2, sm], \(f''''\)[z] \[Rule] 
                    dersam[4, 
                      sm]}\) /. {subst}; \[IndentingNewLine]virfict = \
{f\_\(-2\) \[Rule] 6  f\_0 - 8  f\_1 + 3  f\_2, 
              f\_\(-1\) \[Rule] 3  f\_0 - 3  f\_1 + f\_2, f\_0 \[Rule] 0, 
              f\_n \[Rule] 0, 
              f\_\(n + 1\) \[Rule] 3  f\_n - 3  f\_\(n - 1\) + f\_\(n - 2\), 
              f\_\(n + 2\) \[Rule] 
                6  f\_n - 8  f\_\(n - 1\) + 
                  3  f\_\(n - 2\)}; \[IndentingNewLine]fict = 
            unkfict /. 
              virfict; \[IndentingNewLine]rhs = \(Coefficient[#, 
                    unkmain] &\) /@ \((Table[rhs, {i, 1, n - 1}] /. 
                    virfict)\)/\[Omega]; \[IndentingNewLine] (*\(Print[
                rhs // TableForm];\)*) \[IndentingNewLine]lhs = \
\(Coefficient[#, unkmain] &\) /@ \((Table[lhs, {i, 1, n - 1}] /. 
                  virfict)\); \[IndentingNewLine] (*Print["\<next:\>"]; \
\[IndentingNewLine]Print[
              lhs // TableForm];*) \[IndentingNewLine] (*\(Return[
                rhs . Inverse[lhs]];\)*) \[IndentingNewLine]Return[{lhs, 
              rhs}];\[IndentingNewLine]];\)\)], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
    \( (*\:041a\:0420\:0421\ \:0441\ \:0434\:0440\:043e\:0431\:043d\:043e\
\:0439\ \:0441\:0435\:0442\:043a\:043e\:0439\ \:0438\ \
\:0446\:0435\:043d\:0442\:0440\:0430\:043b\:044c\:043d\:044b\:043c\:0438\ \
\:0440\:0430\:0437\:043d\:043e\:0441\:0442\:044f\:043c\:0438*) \)], "Input"],

Cell[BoxData[
    \(\(FiniteMatrix4Eigen[n_, sample_:  3, subst___] := 
        Block[{h = H\/n, krseq, krs, unk, unkmain, unkfict, i, sm, s, order, 
            lhs, rhs, \[Omega]}, \[IndentingNewLine]sm = 
            Max[sample, 3]; \[IndentingNewLine]unk = 
            Table[f\_\(i + 1/2\), {i, 0 - \(sm - 1\)\/2, 
                n + \(sm - 1\)\/2 - 1}]; \[IndentingNewLine]unkmain = 
            Table[f\_i, {i, 1/2, n - 1/2}]; \[IndentingNewLine]unkfict = 
            Complement[unk, unkmain]; \[IndentingNewLine]dersam[order_, s_] := 
            Table[f\_j, {j, i - \(s - 1\)\/2, i + \(s - 1\)\/2}] . 
                NDCoefficientList[order, s]/
              h\^order; \[IndentingNewLine]rhs = \(\(-D[
                    eqOZ[\[Omega], k, R], \[Omega]]\) /. {z \[Rule] i\ h, 
                  f[z_] \[Rule] dersam[0, sm], \(f'\)[z_] \[Rule] 
                    dersam[1, sm], \(f''\)[z_] \[Rule] 
                    dersam[2, 
                      sm]}\) /. {subst}; \[IndentingNewLine]lhs = \(Simplify[\
\((eqOZ[\[Omega], k, R] - \[Omega]\ D[
                          eqOZ[\[Omega], k, R], \[Omega]])\)] /. {z -> i\ h, 
                  f[z_] -> dersam[0, sm], \(f'\)[z_] -> 
                    dersam[1, sm], \(f''\)[z_] -> 
                    dersam[2, sm]}\) /. {subst}; \[IndentingNewLine]virfict = 
            First[Solve[
                Thread[{dersam[0, sm - 1] /. i \[Rule] 0, 
                      dersam[0, sm - 1] /. i \[Rule] n} \[Equal] 0], 
                unkfict]]; \[IndentingNewLine]fict = unkfict /. virfict; 
          rhs = \(Coefficient[#, 
                  unkmain] &\) /@ \((Table[rhs, {i, 1/2, n - 1/2}] /. 
                  virfict)\); \[IndentingNewLine]lhs = \(Coefficient[#, 
                  unkmain] &\) /@ \((Table[lhs, {i, 1/2, n - 1/2}] /. 
                  virfict)\); \[IndentingNewLine] (*rhs = \(Coefficient[#, 
                    unk] &\) /@ 
                Join[Table[0, {i, 2}, {j, n + sm - 1}], 
                  Table[rhs, {i, 1/2, n - 1/2}]/\[Omega], 
                  Table[0, {i, 2}, {j, 
                      n + sm - 
                        1}]]; \[IndentingNewLine]lhs = \(Coefficient[#, 
                    unk] &\) /@ 
                Join[{dersam[0, sm - 1] /. i \[Rule] 0, 
                    dersam[1, sm - 1] /. i \[Rule] 0}, 
                  Table[lhs, {i, 1/2, n - 1/2}], {dersam[1, sm - 1] /. 
                      i \[Rule] n, 
                    dersam[0, sm - 1] /. 
                      i \[Rule] n}];*) \[IndentingNewLine] (*\(Return[
                rhs . Inverse[lhs]];\)*) \[IndentingNewLine]Return[{lhs, 
              rhs}];\[IndentingNewLine]];\)\)], "Input"]
}, Closed]],

Cell[BoxData[
    \(\(GetIncrements[n_, k1_, R1_] := 
        Eigenvalues[\(Inverse[Last[#]] . First[#] &\)[
            N[FiniteMatrix4Eigen[n, 3, k \[Rule] k1, 
                R \[Rule] R1]]]];\)\)], "Input"],

Cell[BoxData[
    \(\(GetIncrements[n_, k1_, R1_] := 
        Select[Eigenvalues[
            N[FiniteMatrix4Eigen[n, 3, k \[Rule] k1, R \[Rule] R1]]], 
          NumericQ];\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Take[Sort[GetIncrements[20, 1, 1000], Im[#1] < Im[#2] &], 
      10]\)], "Input"],

Cell[BoxData[
    \({\(\(0.9782730066470927`\)\(\[InvisibleSpace]\)\) + 
        0.02337019820915145`\ \[ImaginaryI], \(\(0.9361194037712741`\)\(\
\[InvisibleSpace]\)\) + 
        0.06817258092990564`\ \[ImaginaryI], \(\(0.8966407185944497`\)\(\
\[InvisibleSpace]\)\) + 
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\[InvisibleSpace]\)\) + 
        0.15962955171006454`\ \[ImaginaryI], \(\(0.29991580644528937`\)\(\
\[InvisibleSpace]\)\) + 
        0.1686674666016106`\ \[ImaginaryI], \(\(0.29991580645130717`\)\(\
\[InvisibleSpace]\)\) + 
        0.16866746660468526`\ \[ImaginaryI], \(\(0.8172756510632816`\)\(\
\[InvisibleSpace]\)\) + 
        0.19217326494449902`\ \[ImaginaryI], \(\(0.589215342453095`\)\(\
\[InvisibleSpace]\)\) + 
        0.20068650743919256`\ \[ImaginaryI], \(\(0.5892153499752092`\)\(\
\[InvisibleSpace]\)\) + 
        0.20068757877940863`\ \[ImaginaryI], \(\(0.713618532740849`\)\(\
\[InvisibleSpace]\)\) + 0.20091339478880113`\ \[ImaginaryI]}\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(GetMainIncrement[n_, k1_, R1_, num_:  1] := \((Print[{n, k1, R1}]; 
          Take[Sort[Im /@ GetIncrements[n, k1, R1]], {num}] // 
            First)\);\)\)], "Input"],

Cell[BoxData[
    \(\(GetEigenWave[n1_, k1_, R1_, mode_:  1] := 
        Block[{eigval, eigvec, nom, syst}, 
          syst = \(Inverse[Last[#]] . First[#] &\)[
              N[FiniteMatrix4Eigen[n1, 3, k \[Rule] k1, 
                  R \[Rule] R1]]]; \[IndentingNewLine]{eigval, eigvec} = 
            Select[Transpose[Eigensystem[syst]], NumericQ[First[#]] &] // 
              Transpose; \[IndentingNewLine]nom = 
            Flatten[Position[
                  eigval, \(Sort[eigval, 
                      Im[#1] < Im[#2] &]\)\[LeftDoubleBracket]
                    mode\[RightDoubleBracket]]] // 
              First; \[IndentingNewLine]Print["\<Main eigen value is \>", 
            eigval[\([nom]\)]]; \[IndentingNewLine]Return[
            eigvec[\([nom]\)]];];\)\)], "Input"],

Cell[BoxData[
    \(\(GetEigenWave[n1_, k1_, R1_, mode_:  1] := 
        Block[{eigval, eigvec, nom, syst}, 
          syst = N[
              FiniteMatrix4Eigen[n1, 3, k \[Rule] k1, 
                R \[Rule] R1]]; \[IndentingNewLine]{eigval, eigvec} = 
            Select[Transpose[Eigensystem[syst]], NumericQ[First[#]] &] // 
              Transpose; \[IndentingNewLine]nom = 
            Flatten[Position[
                  eigval, \(Sort[eigval, 
                      Im[#1] < Im[#2] &]\)\[LeftDoubleBracket]
                    mode\[RightDoubleBracket]]] // 
              First; \[IndentingNewLine]Print["\<Main eigen value is \>", 
            eigval[\([nom]\)]]; \[IndentingNewLine]Return[
            eigvec[\([nom]\)]];];\)\)], "Input"],

Cell[BoxData[
    \( (*\:041e\:0442\:043a\:043b\:043e\:043d\:0435\:043d\:0438\:044f\ \:0432\
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\:043d\:043e\:0433\:043e\ \:0437\:043d\:0430\:0447\:0435\:043d\:0438\:044f\
\[IndentingNewLine]With[{n2 = 10, k2 = 1, R2 = 10000, 
            mod = 1}, \(Det[
                SetPrecision[
                  First[#] - \(Sort[GetIncrements[n2, k2, R2], 
                          Im[#1] < Im[#2] &]\)[\([mod]\)] Last[#], 1000]] &\)[
            SetPrecision[
              FiniteMatrix4Eigen[n2, 3, k \[Rule] k2, R \[Rule] R2], 
              1000]]]*) \)], "Input"],

Cell[BoxData[
    \(\(\( (*\:041e\:0442\:043a\:043b\:043e\:043d\:0435\:043d\:0438\:044f\ \
\:0432\ \:0432\:0435\:043a\:0442\:043e\:0440\:0435\ \:043f\:043e\:0441\:043b\
\:0435\ \:043f\:043e\:0434\:0441\:0442\:0430\:043d\:043e\:0432\:043a\:0438\ \
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\:043d\:0430\:0447\:0435\:043d\:0438\:0439\ \:0438\ \:0432\:0435\:043a\:0442\
\:043e\:0440\:043e\:0432*) \)\(\[IndentingNewLine]\)\(\(With[{n2 = 260, 
              k2 = 1, R2 = 1000, 
              mod = 1}, \(\((First[#] - \(Sort[GetIncrements[n2, k2, R2], 
                            Im[#1] < Im[#2] &]\)[\([mod]\)] Last[#])\) . 
                  GetEigenWave[n2, k2, R2, mod] &\)[
              FiniteMatrix4Eigen[n2, 3, k \[Rule] k2, R \[Rule] R2]]] // 
          Abs\) // PrintRange;\)\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(incrs = 
        Sort[GetIncrements[300, 1, 10000] // N, 
          Im[#1] > Im[#2] &];\)\), "\[IndentingNewLine]", 
    \(\(Print[Min[Im /@ incrs]];\)\), "\[IndentingNewLine]", 
    \(\(Show[
        Graphics[{PointSize[0.04], \(Point[{Re[#], Im[#]}] &\) /@ incrs}], 
        Axes \[Rule] True, 
        PlotRange \[Rule] {All, {\(-1\), 5}}];\)\)}], "Input"],

Cell[BoxData[
    \(0.0071710683581145684`\)], "Print"],

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